A Simple Model for Nonlinear Critical Layers in an Unstable Baroclinic Wave

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  • 1 Woods Hole Oceanographic Institution, Woods Hole, MA 02543
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Abstract

Weakly nonlinear theory is developed for finite-amplitude dynamics of a slightly dissipative baroclinic wave at the point of minimum critical shear in the β-plane two-layer model. At this parameter setting the nonlinear theory provides a simple manifestation of critical layer dynamics since the Doppler-shifted frequency vanishes in one of the two layers. Calculations show that when the dissipation is proportional to the potential vorticity and is weak, the new equilibrium steady state has uniform potential vorticity in the critical layer although this is not required for wave stabilization. The spatial harmonics of the fundamental play an important role in both the transient and final state. For a weakly dissipative flow, the potential vorticity due to the harmonics is conserved along streamlines of the fundamental wave. An analytical theory is given for the equilibrated wave amplitude based on the assumption of uniform potential vorticity in the critical layer, and this prediction agrees well with the calculations.

Potential vorticity smoothing does not occur when either dissipation or time dependence becomes large.

Abstract

Weakly nonlinear theory is developed for finite-amplitude dynamics of a slightly dissipative baroclinic wave at the point of minimum critical shear in the β-plane two-layer model. At this parameter setting the nonlinear theory provides a simple manifestation of critical layer dynamics since the Doppler-shifted frequency vanishes in one of the two layers. Calculations show that when the dissipation is proportional to the potential vorticity and is weak, the new equilibrium steady state has uniform potential vorticity in the critical layer although this is not required for wave stabilization. The spatial harmonics of the fundamental play an important role in both the transient and final state. For a weakly dissipative flow, the potential vorticity due to the harmonics is conserved along streamlines of the fundamental wave. An analytical theory is given for the equilibrated wave amplitude based on the assumption of uniform potential vorticity in the critical layer, and this prediction agrees well with the calculations.

Potential vorticity smoothing does not occur when either dissipation or time dependence becomes large.

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